Infinite tetration and superroot of infinitesimal

[andydude]

First, if I were you, I would not use i for this infinitesimal construction, as it is not related to the imaginary unit at all. I would use z instead, to be clear, and since the numbers are close to zero. Second, it sounds like your definition of infinitesimals are equivalent to the reciprocals of ordinal numbers, since you talk about \( 0 < z < \mathbb{R}^+ \) (mirroring \( \omega > \mathbb{N} \)) and "infinities of the same order", so you might be better off, and able to communicate your ideas better if you simply define your transfinitesimals in terms of Cantor's ordinal numbers instead. So you could define your infinitesimals as: \( z := \frac{1}{\omega} \) in terms of the first transfinite ordinal number \( \omega \).

My only problem with this is that I have a deep-seated hatred of Cantor's numbers. I'll save that talk for later.

Andrew Robbins